In trigle ABC, the points D, E, F are the midpoints of the sides BC,CA, and AB respee. Tively. Using vector methed ,show that the area of ` Delta ` DEF is equal to ` (1)/(4) ` (area of ABC).

If D, E and F be the middle points of the sides BC,CA and AB of the DeltaABC , then AD+BE+CF is

In a triangle ABC, if D and E are mid points of sides AB and AC respectively. Show that vecBE+ vecDC=(3)/(2)vecBC .

Let A, B, and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that vec(AD)+vec(BE)+vec(CF)=vec(0) .

From a point O inside a triangle ABC, perpendiculars OD, OE and OF are drawn to the sides BC, CA and AB, respectively. Prove that the perpendiculars from A, B and C to the sides EF, FD and DE are concurrent

Let A(p^2,-p),B(q^2, q),C(r^2,-r) be the vertices of triangle ABC. A parallelogram AFDE is drawn with D,E, and F on the line segments BC, CA and AB, respectively. Using calculus, show that the maximum area of such a parallelogram is 1/2(p+q)(q+r)(p-r)dot

If D and E, are the midpoints of the sides AB and AC of a triangle ABC, prove that vec(BE) +vec(DC)=(3)/(2)vec(BC).

P and Q are the mid-points of the sides CA and CB respectively of a Delta ABC , right angled at C. Prove that 4(AQ^(2)+BP^(2))=5 AB^(2) .

G is the centroid of triangle A B Ca n dA_1a n dB_1 are rthe midpoints of sides A Ba n dA C , respectively. If "Delta"_1 is the area of quadrilateral G A_1A B_1a n d"Delta" is the area of triangle A B C , then "Delta"//"Delta"_1 is equal to a. 3/2 b. 3 c. 1/3 d. none of these

In triangle ABC, if P, Q, R divides sides BC, AC, and AB, respectively, in the raito k : 1 (in order). If the ratio (("area "DeltaPQR)/("area " DeltaABC)) " is " (1)/(3) , then k is equal to